For me it's like this: If the team is rebuilding, then yes, they'll throw the veterans much earlier than they naturally would. That was/is happening to us. It began with the middle of the OL. But if the GM/HC has his job on the line, he'll resort to veteran players.

A rebuilding team takes time to win, and veteran, accomplished players usually want to win now, so when the team don't make moves to win in the current season, they "thrashtalk" the GM and Coaches. It's natural.

One can make an argument about the fear of the veterans demeaning the leadership, but I don't think it's the age issue. Although it's a lot easier to command respect from someone younger than at your age, but if you're competent and have accomplished something, people will respect you no matter your age.

It's happening more nowadays than it used to IMHO because of the free agency...

Unless a player is insubordinate and disruptive I don't think most coaches or execs mind receiving helpful in-put from veterans or overlook the value of their experience and leadership.
I suspect it has more to do with the wear and tear and increased risk of investment to injury with older players than with younger ones.
We are coming to the end of a decade period, which is usually when teams and the league undergo most significant turnover. I suspect if we looked at vets being let go and not actively pursued now with 1999, 1989, 1979, etc. we would find a similar pattern.